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Home » 807: The Secret Code You’ve Been Searching For (Explained!) - Abbey Badges

807: The Secret Code You’ve Been Searching For (Explained!) - Abbey Badges

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 + m = 2(-5 + 3m)

📌 + m = -10 + 6m

📌 = 5m \Rightarrow m = 1

📌 The value of $ m $ is $ \boxed{1} $.**Question:** A box contains 8 red marbles, 6 blue marbles, and 4 green marbles. How many ways can 5 marbles be selected if at least one marble of each color must be included?

📌 To ensure that at least one marble of each color is selected, we must consider all valid combinations of red (R), blue (B), and green (G) marbles such that:

📌 $ r \geq 1 $, $ b \geq 1 $, $ g \geq 1 $

📌 $ r + b + g = 5 $

📌 $ r \leq 8 $, $ b \leq 6 $, $ g \leq 4 $

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a₁ = 1.2 → a(1)² + b(1) + c = a + b + c = 1.2
a₂ = 1.8 → 4a + 2b + c = 1.8
a₃ = 2.5 → 9a + 3b + c = 2.5
Subtract first from second: (4a + 2b + c) − (a + b + c) = 1.8 − 1.2 →...
Subtract second from third: (9a + 3b + c) − (4a + 2b + c) = 2.5 − 1.8...
Now subtract: (5a + b) − (3a + b) = 0.7 − 0.6 → 2a = 0.1 → a = 0.05
Then 3(0.05) + b = 0.6 → 0.15 + b = 0.6 → b = 0.45
Then a + b + c = 1.2 → 0.05 + 0.45 + c = 1.2 → c = 0.7
So aₙ = 0.05n² + 0.45n + 0.7
a₄ = 0.05(16) + 0.45(4) + 0.7 = 0.8 + 1.8 + 0.7 = <<0.8+1.8+0.7=3.3>>3...

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